1. Field of Invention:
The present invention relates to obtaining measures of parameters of interest concerning possible petroleum content of subsurface formations.
2. Description of the Prior Art:
In assessing possible petroleum deposits in a subsurface formation adjacent a well borehole, certain parameters of the formation have long been of particular interest. They typically include water saturation, both of the uninvaded formation and the flushed zone adjacent the borehole; shale (or other rock) volume fraction of the formation; porosity of the formation; and hydrogen index of the formation constituents. None of these parameters, however, is directly measurable, so far as is known. A number of types of well logging tools and methods exist for obtaining responses of the formation to the particular logging tool used. Several parameters of interest typically contribute to each of the various types of log measurements obtained, usually in a non-linear fashion. Further, the contribution of a parameter of interest is often of a different nature in different types of logs obtained.
Conventional formation evaluation methods generally relied on "multiple pass" solutions which were inherently piecemeal. These methods were not self-consistent, which frequently led to incorrect solutions, such as water saturations greater than 100 percent or negative porosities. The conventional methods are well-known and have been extensively discussed in the literature.
U.S. Pat. No. 4,338,664 represented an attempt to determine the value of parameters of interest form a set of logging measurements. The method described in this patent relied on a computation technique which introduced an incoherence function, G(x), which was defined by: ##EQU1## where F(x) is an error function representing the sum of differences between the calculated data values and the actually measured values of the data, divided by a weighting factor. In Equation (1), the terms in the summation were termed penalty functions which are added to impose, for example, constraints of the form: EQU g.sub.k (x).gtoreq.0 (2)
The parameters .tau..sub.k in this penalty function are auxiliary parameters which were not well-defined. That is, there was no systematic procedure for assigning values to the .tau..sub.k, which was one of the disadvantages of this method. The function g.sub.k (x) was set equal to zero if the constraint was obeyed and g.sub.k =g.sub.k if the constraint was not obeyed.
The method of this patent thus used what is known in non-linear programming as an unconstrained minimization method (e.g., Fletcher-Powell) to obtain the minimum of the auxiliary function G(x) with respect to x (e.g., the vector x denotes a set of reservoir properties).
Let x denote the value of x which minimizes G(x). In the penalty function method, it can be proven that: ##EQU2## for .tau..sub.1, .tau..sub.2, . . . .tau..sub.n, where x* is the value of x which minimizes F(x). In order to properly apply the penalty function method, however, the procedures outlined below had to be followed:
(1) Select a set a values .tau..sub.k and an initial guess for the solution x.sub.1 , PA0 (2) Find x using the Fletcher-Powell minimization method, PA0 (3) Reduce the values of .tau..sub.k and return to step 2 using x as the initial guess. Continue this process until reduction of .tau..sub.k does not significantly alter the value of x . Then, using Equation (3), one could accept x as an accurate estimate of x*.
The procedure described above was inefficient in that it required repeated application of the unconstrained minimization algorithm. The inefficiency resulted from the fact that convergence with respect to the .tau..sub.k had to be achieved, obtaining the minimum of the error function F(x) by an indirect and inefficient procedure.